List of practice Questions

If force \( \vec{F}= 2t\,\hat{i} + 3t^2\,\hat{j} \) acts on a particle of mass \(m = 2\,\text{kg}\), find the power at \(t = 2\,\text{s}\) if the particle was initially at rest.
An ideal gas has number of moles \(n=2\), initial volume \(V_0\) and pressure \[ P=P_0\left[1+\left(\frac{V_0}{V}\right)^2\right]^{-1}. \] The gas goes from state \(A\) (initial) to \(B\) (final) such that volume becomes \(3V_0\). Find \(T_A-T_B\).
If $A = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}$ and $B = \tan 81^\circ - \tan 3^\circ$, find $\frac{B}{A}$
If $\vec{a}_R = \tan \theta_R \hat{i} + \hat{j}$ and $\vec{b}_R = \hat{i} - \cot \theta_R \hat{k}$ where $\theta_R = \frac{2^{R-1} \pi}{2^N + 1}$, then the value of $\frac{\sum_{R=1}^N |\vec{a}_R|^2}{\sum_{R=1}^N |\vec{b}_R|^2}$ is
If $f(x) = \min \{2x^2 + 3, 6x\} + |x-1| \cos(x^2 - \frac{1}{4})$, then the number of points of non derivability of $f(x)$ is/are
If $\int_{-2}^2 ([\sin x] + |x \sin x|) dx = 2 \sin 2 - 4 \cos 2 - \beta$, then the value of $|\beta|$ where $[\cdot]$ is GIF is
Consider the differential equation $\frac{dy}{dx} = (x^2 + x + 1)(y^2 - y + 1)$. If $y(0) = \frac{1}{2}$, then the value of $2(y(1)) - 1$ is
The domain of $f(x) = \cos^{-1} \left( \frac{4x + 2[x]}{3} \right)$ (where $[x]$ denotes greatest integer function) is
If $Z$ be a complex number such that $|Z + 2| = |Z - 2|$ and $\text{arg} \left( \frac{Z - 3}{Z + 1} \right) = \frac{\pi}{4}$, then the value of $|Z|^2$ is
If coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}$ is $\binom{100}{3} \left( \frac{101}{4} - 43n \right)$, then the value of $(k^3 + 43n)$ is
Let $y = \tan^{-1}\left( \frac{3\cos x - 4\sin x}{4\cos x + 3\sin x} \right) + \tan^{-1}\left( \frac{x}{1 + \sqrt{1 + x^2}} \right)$. Then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$ is :
Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{a, b, c\}$ then total number of functions from $A$ to $B$ which are not onto are
If Latus rectum of parabola $y^2 = 4kx$ and ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ coincide then the value of $e^2 + 2\sqrt{2}$ is, where $e$ is eccentricity of ellipse
In an A.P. first term is $\frac{10}{3}$ and first 30 terms are non-negative such that sum of first 30 terms = $(T_{30})^3$, then d is equal to (where $T_n$ is $n^{\text{th}}$ term of A.P., and d is the common difference of A.P.)
If $f(x)$ is a non-constant polynomial satisfying $f(x) = f'(x)f''(x)$ and $f(0) = 0$. Then the value of $\int_0^2 f(x) dx + f'(2) + f''(2)$ is :
Let in a $\triangle ABC$, given that $A \equiv (1, 2)$, mid-point of $AB$ is $(-5, -1)$ and centroid is $(3, 4)$ then circumcentre is $(\alpha, \beta)$, then the value of $21(\alpha + \beta)$ is :
Let mean and median of 9 observations 8, 13, a, 17, 21, 51, 103, b, 67 are 40 and 21 respectively where a > b. If mean deviation about median is 26 then 2a is :-
A circle $x^2 + y^2 + x - 3y = 0$ passes through $P(1, 2)$. If 2 chords (PS \& PR) drawn from P are bisected by $y$-axis, then mid point of RS is $(\alpha, \beta)$, find $6(\alpha + \beta)$
Find number of ways of arranging 4 Boys \& 3 Girls such that all girls are not together :
If $S = \{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, A^2 - 4A + 3I = \text{Null matrix}, a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } \text{Tr}(A) = 4 \}$. Then the number of elements of set $S$ is/are
Match the column-I with column-II:
Choose the correct match.
IUPAC Name of the formed compound:
Consider the following reaction:
What is the colour of the final compound B?
Let $A$ is a matrix of order 3 such that $|A| = -4$, then the value of $|\text{adj}(\text{adj}(2\text{adj} A)^{-1})|$ is
Choose the correct match.